In this paper, we prove some Milne type inequalities for interval-valued functions and, along with it, we explore some connections with other inequalities. More precisely, using the Aumann integral and the Kulisch–Miranker order and including-order on the space of real and compact intervals, we establish some Milne type inequalities for interval-valued functions. Also, using different orders, we obtain some connections with Chebyshev, Cauchy–Schwarz, and Hölder inequality. Finally, some new ideas and results based on submodular measures are explored as well as some examples and applications are presented for illustrating our results.

在本文中，我们证明了区间值函数的一些Milne型不等式，并与之一起探讨了与其他不等式的一些联系。更精确地，使用Aumann积分和Kulisch-Miranker阶以及实和紧区间的空间上的包含阶，我们为区间值函数建立了一些Milne型不等式。同样，使用不同的阶数，我们获得了与切比雪夫（Chebyshev），柯西·施瓦茨（Cauchy-Schwarz）和霍尔德不等式的联系。最后，探讨了基于亚模量测的一些新思想和新结果，并给出了一些实例和应用来说明我们的结果。